Entropy stability analysis of smoothed dissipative particle dynamics

TL;DR

An entropy stability analysis of smoothed dissipative particle dynamics (SDPD) is presented to review the validity of particle discretization of entropy equations and suggests that there exist eight different types of entropy stability conditions, which depend on the types of kernel functions.

Abstract

This article presents an entropy stability analysis of smoothed dissipative particle dynamics (SDPD) to review the validity of particle discretization of entropy equations. First, we consider the simplest SDPD system: a simulation of incompressible flows using an explicit time integration scheme, assuming a quasi-static scenario with constant volume, constant number of particles, and infinitesimal time shift. Next, we derive a form of entropy from the discretized entropy equation of SDPD by integrating it with respect to time. We then examine the properties of a two-particle system for a constant temperature gradient. Interestingly, our theoretical analysis suggests that there exist eight different types of entropy stability conditions, which depend on the types of kernel functions. It is found that the Lucy kernel, poly6 kernel, and spiky kernel produce the same types of entropy stability conditions, whereas the spline kernel produces different types of entropy stability conditions. Our results contribute to a deeper understanding of particle discretization.

Figures (6)

• Figure 1: Function of $\Lambda(r)$.
• Figure 2: Fourth-order function of $f(b,C)$ described by Eq. (\ref{['eq:4thorderfunc']}).
• Figure 3: Dependencies of $X_1$ and $X_{2}$ on the parameter $\lambda$.
• Figure 4: Classification of the state of the SDPD system determined by the parameters $b$ and $\lambda$ when $f(b,C)\ge0$.
• Figure 5: Comparison of function $F$ in different types of kernel functions.